طراحی بهینه از دستگاه های الکترومغناطیسی: توسعه یک ابزار بهینه سازی کارآمد بر اساس اجرا عملیات جهش های هوشمند در یک الگوریتم ژنتیک

Topology optimization methods are aimed to produce optimal design. These tools implement optimization algorithms that modify the distribution of some materials within a predefined design space without a priori ideas regarding the topology or the geometry of the best solution. In this paper, we study a specific tool that combines a genetic algorithm, a material distribution formalism based on Voronoi cells and a commercial FEM evaluation tool. In particular, this paper shows, through a simple but representative case study, that it is possible to improve the performance of the topology optimization tool during the local search phase, i.e. the geometrical and dimensional optimization phase for which the topology optimization methods are originally not well-suited.

Today, optimization methods are becoming increasingly important in the design process of devices, whether electromagnetic or other [23], [4] and [24]. These methods can help the designers in different ways in the design process. Actually, three approaches can be distinguished. Parametric optimization first, plays on the dimensional parameters of a solution whose geometry, and therefore topology, are defined a priori and once and for all the optimization process by the designer [5], [19] and [1]. Geometric or shape optimization expands the space of possible solutions by changing the dimensions but also the geometry of a solution whose topology is defined by the designer [15] and [10]. Topology optimization finally, opens further the space of possible solutions by searching the best way to distribute given materials in a design space according to one or more objective functions [2], [3], [18] and [11]. The final solution is therefore free of any initial geometry or topology proposed by the designer. Mostly topology optimization tools involve three coupled modules: an optimization algorithm, a material distribution formalism and an evaluation tool. The optimization algorithm deals with the design parameters with the aim to maximize or minimize an objective function. These design parameters are used by the material distribution formalism. This module is necessary to link the set of design parameters and the graphical representation of the solution. Then, the graphical description is transferred to the evaluation tool. This last module plays two roles. The first consists in producing a numerical model of the solution described through the material distribution formalism and the design parameters, and in solving it. The second consists in computing the objective functions and the constraints. Optimization algorithms are numerous. However, as the design problems treated by the topology optimization methods generally involve simultaneously a large number of design variables and nonlinear and nonconvex objective functions and constraints, the choice of the optimization algorithm as well as its implementation are decisive. The literature in the field of topology optimization shows that the optimization algorithms used are as well deterministic algorithms [2], [16], [17] and [13] as metaheuristic algorithms [21] and [7]. In the former case, the information of the gradient of the objective function is used to indicate the direction in which to change the design variables. Although offering a faster convergence, these methods handle difficult the nonconvex nature of the problems addressed [13]. In the second case, the stochastic nature of the algorithms avoids the trap of local minimum but at the cost of much slower convergence. This paper focuses on metaheuristic algorithms and especially on genetic algorithms. Despite their ability to find the optimal topology, topology optimization tools using metaheuristic algorithms have some difficulties to converge both in terms of geometry and in terms of dimension. In order to obtain a final design, the user must therefore use additional tools, like geometric or parametric optimization tools, to optimize the solution with the topology found by the topology optimization tool [4]. To overcome this problem of convergence, some papers propose new operators for the optimization algorithm [6], [12] and [14] or the material distribution formalism [22]. The aim of this paper is to present a study of new adapted mutations methods for the genetic algorithm coupled with the material distribution formalism, namely Voronoi diagram, in order to improve the performance of the topology optimization tool in the local search phase necessary to refine the geometry and the dimension of the final solution. This paper is divided into four sections. Section 2 describes the reference topology optimization tool. Section 3 presents a study of new adapted mutation operators with a Voronoi formalism in order to improve local search step. The study case used as benchmark is described in Section 4. Finally, the results are exposed and analyzed in Section 5 while Section 6 is devoted to the conclusion.

Genetic algorithms, like many metaheuristics have a very good global search capability, but they are less suitable for local search. Used in topology optimization tools, these algorithms are therefore very effective at finding the optimal topology but less to refine the geometry or dimensions of the solution. In this paper, we showed that it is possible to adapt the genetic algorithm by introducing a specific mutation mechanism so as to improve its ability to perform dimensional and geometric optimization. This mechanism is based on the possibility offered by the material distribution formalism, in this instance the Voronoi diagram, to modify the subdivision of the design space by moving the Voronoi center. It consists then in changing the amplitude of movement, characterized by a parameter α, during the optimization process in order to support the global search at an early stage of the optimization and the local search afterwards. Three different models of evolution of α are presented and compared to a static model considering α constant throughout the optimization. The first is changing linearly α depending on the number of generations. The second is based on the average minimum distance between cells. The third set independently for each cell a value of α directly related to the distance between the cell and its nearest neighbor. The comparisons made on a case study derived from an electromagnetic inverse problem shows that the three models proposed do not change the convergence of the algorithm in the global search phase. However, they offer a better convergence than the static model in the local search phase. This results in a final solution whose geometry is actually better, even if there is no gain in the topology.